Chapter 12: Partial Derivatives
Section 12.1: Functions of Several Variables
Definition: A function of two variables is a rule that assigns a unique real number
f (x, y) to each ordered pair (x, y) ∈ D ⊂ R2 . The set D is called the domain of f and
{f (x, y)|(x, y) ∈ D} is called the range of f .
Example: Find and sketch the domain of the given functions:
p
(a) f (x, y) = x2 + y 2 − 1 + ln(4 − x2 − y 2 )
The function is defined for x2 + y 2 − 1 ≥ 0 and 4 − x2 − y 2 ≥ 0. The domain is
D = {(x, y) ∈ R2 |1 ≤ x2 + y 2 < 4}.
This is an annular region in the xy-plane.
(b) f (x, y) =
√
x+
√
y+
p
x2 + y 2 − 16
The function is defined for x ≥ 0, y ≥ 0, and x2 + y 2 − 16 ≥ 0. The domain is
D = {(x, y) ∈ R2 |x ≥ 0, y ≥ 0, x2 + y 2 ≥ 16}.
This is the portion of the region outside the circle centered at the origin with radius 4
in the first quadrant.
p
9 − x2 − y 2
(c) f (x, y) =
2x + y
The function is defined for 9 − x2 − y 2 ≥ 0 and y =6= −2x. The domain is
D = {(x, y) ∈ R2 |x2 + y 2 ≤ 9, y 6= −2x}.
This is the portion of the disk centered at the origin with radius 3 such that y 6= −2x.
Definition: If f is a function of two variables with domain D, then the graph of f is the set
{(x, y, z) ∈ R3 |z = f (x, y), (x, y) ∈ D}.
Note: The graph of a function f of two variables is a surface with equation z = f (x, y). The
graph of f can be visualized as lying directly above or below the domain D in the xy-plane.
Figure 1: Graph of the surface defined by z = f (x, y) for (x, y) ∈ D.
Example: Sketch the graph of the following functions:
(a) f (x, y) = 3
The surface is the horizontal plane z = 3.
(b) f (x, y) = 4 − 2x − y
The surface is defined by
z = 4 − 2x − y
2x + y + z = 4,
which is the equation of a plane.
(c) f (x, y) = x2 + 9y 2
The surface is defined by z = x2 + 9y 2 , which is an elliptic paraboloid.
Definition: The level curves of a function of two variables are the curves defined by f (x, y) =
k, where k is a constant in the range of f . The level curves of f (x, y) are the horizontal
traces of the graph of f in the plane z = k projected onto the xy-plane. A graph of the level
curves is called a contour plot.
Figure 2: Level curves and contour plot of a surface z = f (x, y).
Note: Contour plots are commonly used in topographic maps.
Example: Sketch the level curves for the given functions:
(a) f (x, y) = x2 + 4y 2 for k = 0, 4, 16
If k = 0, the equation x2 + 4y 2 = 0 defines the point (0, 0). If k = 4, the equation
x2 + 4y 2 = 4 defines an ellipse
x2
+ y 2 = 1.
4
If k = 16, the equation x2 + 4y 2 = 16 defines an ellipse
x2 y 2
+
= 1.
16
4
The level curves are ellipses.
Figure 3: Contour plot for f (x, y) = x2 + 4y 2 .
(b) f (x, y) = y − x2 for k = −1, 0, 1, 2
The level curves y − x2 = k are parabolas
y = x2 + k,
k = −1, 0, 1.
Figure 4: Contour plot for f (x, y) = y − x2 .
(c) f (x, y) =
p
4 − x2 − y 2 for k = 0, 1, 2
The level curves
p
4 − x2 − y 2 are circles
p
4 − x2 − y 2 = k
4 − x2 − y 2 = k 2
x2 + y 2 = 4 − k 2 .
Figure 5: Contour plot for f (x, y) =
p
4 − x2 − y 2 .